Matroids And Greedy Algorithms. A Deeper Justification of Using Greedy Approach To Find A Maximal set of a Matroid

Greedy algorithms are used in solving a diverse set of problems in small computation time. However, for solving problems using greedy approach, it must be proved that the greedy strategy applies. The greedy approach relies on selection of optimal choice at a local level reducing the problem to a single sub problem, which actually leads to a globally optimal solution. Finding a maximal set from the independent set of a matroid M(S, I) also uses greedy approach and justification is also provided in standard literature (e.g. Introduction to Algorithms by Cormen et .al.). However, the justification does not clearly explain the equivalence of using greedy algorithm and contraction of M by the selected element. This paper thus attempts to give a lucid explanation of the fact that the greedy algorithm is equivalent to reducing the Matroid into its contraction by selected element. This approach also provides motivation for research on the selection of the test used in algorithm which might lead to smaller computation time of the algorithm

Polymatroid greedoids

AbstractThis paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated

A Note on Selectors and Greedoids

This note deals with relations between selectors studied by Henry Crapo and a special class of greedoids introduced by these authors in a previous paper. We show that selectors are greedoids with the interval property and that a second property of Crapo, which he calls ‘locally free’ is the interval property without upper bound. In the last section of the paper we show that retract sequences of posets are general greedoids, but not selectors

A combinatorial characterization of S2S_2 binomial edge ideals

Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut-point sets of a graph $G$, special sets of vertices that disconnect $G$ in a minimal way, play an important role since they are in bijection with the minimal prime ideals of $J_G$. In this paper we establish the first graph-theoretical characterization of binomial edge ideals $J_G$ satisfying Serre's condition $(S_2)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point sets of $G$ form an accessible set system. The proof relies on the combinatorial structure of the Stanley-Reisner simplicial complex of a multigraded generic initial ideal of $J_G$, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent

Cohen-Macaulay binomial edge ideals of small graphs

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal JG to special disconnecting sets of vertices of its underlying graph G, called cut sets. More precisely, the conjecture states that JG is Cohen-Macaulay if and only if JG is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results

Das Twisten von Matroiden

Das Twisten von Matroiden steht für das Versammeln der symmetrischen Differenzen aller Matroid-Mengen mit einer vorgegebenen Teilmenge der Grundmenge, die Matroide dabei im Kontext der Mengensysteme interpretierend. In dieser Arbeit werden die so entstehenden Mengensysteme eingehend untersucht, samt einiger Derivate und beschreibender Funktionen. Ein Schwerpunkt besteht in der Einordnung der getwisteten Matroide in ein Gefüge von bekannten Greedoid-Klassen, beschränkt hier auf Systeme, die auf ungeordneten Mengen basieren und eine Affinität zu Greedy-Algorithmen bezüglich linearer Optimierung aufweisen. Diese Beziehungen werden beschrieben und die Klassen voneinander abgegrenzt. Zusätzlich wird eine Greedoid-Eigenschaft hervorgehoben, die die Bildung einer weiteren Klasse rechtfertigen soll, mit der Begründung, dass diese Systeme, falls sie gleichzeitig Delta-Matroide sind, die lineare Optimierung einem hier dargelegten Greedy-Algorithmus anvertrauen dürfen. Dieser benötigt lediglich das gewöhnliche Mengensystem-Orakel, um in polynomiell vielen Zeitschritten, abhängig von der Größe der Grundmenge, erfolgreich zu sein. In diese Klasse gehören neben den getwisteten Matroiden auch die Matroid-Twistvereinigungen, die hier vorgestellt und diesbezüglich untersucht werden. Das Auftreten dieser Konstrukte im Rahmen des Traveling-Salesman-Problems und des verwandten Vehicle-Routing-Problems wird beschrieben. Ein Exkurs in die Polyedertheorie beinhaltet den Nachweis, dass der von Dunstan und Welsh vorgestellte verallgemeinerte Polymatroid-Algorithmus eine Charakterisierung bisubmodularer Polyeder bereitstellt. Dabei kommt der Spiegelung, als Vektorraum-Analogon zum Twisten, eine Funktion zu, die auch zu weiteren Beschreibungen für Delta-Matroide führt